Colin MacLaurin

this principle seems to be inconsistent with... admitting... an infinitely little quantity or difference, which, added to itself any number of times, is never supposed to become equal to any finite quantity whatsoever.

They found, that similar triangles are to each other in the duplicate ratio of their homologous sides; and, by resolving similar polygons into similar triangles, the same proposition was extended to these polygons also. But when they came …

The method of demonstration, which was invented by the author of fluxions, is accurate and elegant; but we propose to begin with one that is somewhat different; which, being less removed from that of the antients, may make the transition …

In delivering the principles of this method, we apprehend it is better to avoid such suppositions: but after these are demonstrated, short and concise ways of speaking, though less accurate, may be permitted, when there is no hazard of our …

We shall not consider any part of space or time as indivisible, or infinitely little; but we shall consider a point as a term or limit of a line, and a moment as a term or limit of time: nor shall we resolve curve lines, or curvilineal …

his conciseness maybe supposed to have given some occasion to the objections which have been raised against his method.

He considered magnitudes as generated by a or motion, and showed how the velocities of the generating motions were to be compared

especially after he saw that this liberty was growing to so great a height. In demonstrating the grounds of the method of fluxion, he avoided them, establishing it in a way more agreeable to the strictness of geometry.

There were some, however, who disliked the... use of infinites and infinitesimals in geometry. Of this number was Sir Isaac Newton

an infinite scale of infinites and s (ascending and descending always by infinite steps) was imagined and proposed to be received into geometry, as of the greatest use for penetrating into its abstruse parts. Some have argued for …

The science being now vastly enlarged, and applied with success to philosophy and the arts, it is of greater importance than ever that its evidence be preserved perfect.

If, upon the whole, the Evidence of this method be represented to the satisfaction of the Reader, some of the abstruse parts illustrated, or any improvements of this useful Art be proposed, I shall be under no great concern, though …

In most of the instances wherein my conclusions did not agree with those given by other Authors, I have not mentioned their names.

Several Treatises have appeared while this was in the press, wherein some of the same Problems have been considered, though generally in a different manner. I have had occasion to mention most of them in the last Chapter of the second …

I have always represented Fluxions of all... Orders by finite Quantities, the Supposition of an infinitely little Magnitude being too bold a Postulatum for such a Science as Geometry.

The greatest part of the first Book was printed in 1737: but it could not have been so useful to the Reader without the second; and I... recommend... to peruse the first Chapters of the second Book, before the five last of the first; there …

I accommodated my Definition of the Variation of Curvature in Chap. xi. to Sir Isaac Newtons, to prevent mistakes, as I have observed in Article 386, but made no material alteration in any thing else.

After I saw that so much had been written upon it to no good purpose, I was rather induced to delay the publication of this Treatise, til I could finish my design.

Besides an answer to '... the Author concealed his real name... a second, by the same hand, in Defense of the first, a Discourse by Mr. Robins, a Treatise of Sir Isaac Newton, with a Commentary by Mr. Colson, and several other Pieces, were …

[T]he Defense of the , and of the great Inventor, was not neglected.

And as this has been the occasion of my delay in publishing... I hope it will serve for an apology, if some mistakes have escaped me in treating of such a variety of subjects, in a manner different from that in which they have usually been …

These, with other observations concerning this method, and its application, led me on gradually to compose a Treatise of a much greater extent than I intended, or would have engaged in, if I had been aware of it when I began this Work, …

I perceived that some Rules were defective or inaccurate; that the Resolution of several Problems which had been deduced in a mysterious manner, by second and third s, could be completed with greater evidence, and less danger of error, by …

This determined me, immediately after that Piece came to my hands, and before I knew any thing of what was intended by others in answer to it, to attempt to deduce those Elements after the manner of the Antients, from a few unexceptionable …